A construction for polynomials.
Dear list, I am looking for a reference to the following construction. In the simplest case, consider the forgetful functor P:Ring->Set, from the category of commutative rings with a unit. Then the set of natural endomorphisms Nat(P, P) can be identified with the set of polynomials of one variable with integer coefficients Z[X]. This can be easily seen by the chain of isomorphisms Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X], where the first isomorphism is due to Z[X] being a free ring with one generator and the second is by Yoneda's lemma. Alternatively, just observe that polynomial functions are precisely ones commuting with every ring homomorphism. In general, let P:C->B be a functor from an arbitrary C to a cartesian closed B. Select an object R in C and let R/P:R/C->B be the "obvious" forgetful functor from the co-slice category. For an object S of B define R[S] = Nat(R/P * hom(S, -), R/P), where hom is the internal hom functor of B and * is functor composition in the diagrammatical order. For C = Ring, B = Set this gives the usual ring of polynomials with coefficients in R and variables from S. This construction extends to a functor C x B -> Set and has some nice properties of the usual polynomials: polynomials of "one variable" (i.e., when S = 1) can be composed, to each r:1->R and x:1->S corresponds a polynomial (provided P maps 1 to 1). Has this or dual (where C/R is used instead of R/C) construction been studied? Maybe in enriched contexts? Thank you, Nikita. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Greetings, I think you've rediscovered, in
Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X],
the unary fragment of what Lawvere called the algebraic structure of that functor Rings --> Sets. The n-ary fragment resides in the analogous calculation Nat (P^n, P) = Nat(Ring(Z[X], -)^n, P) = Nat(Ring(Z[X_1, ..., X_n], -), P) = Z[X_1, ..., X_n]. Cheers, -- Fred ------ Original Message ------ Received: Mon, 04 Apr 2016 08:40:19 PM EDT From: Nikita Danilov <danilov@gmail.com> To: <categories@mta.ca> Subject: categories: A construction for polynomials.
Dear list,
I am looking for a reference to the following construction.
In the simplest case, consider the forgetful functor P:Ring->Set, from the category of commutative rings with a unit. Then the set of natural endomorphisms Nat(P, P) can be identified with the set of polynomials of one variable with integer coefficients Z[X]. This can be easily seen by the chain of isomorphisms
Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X],
where the first isomorphism is due to Z[X] being a free ring with one generator and the second is by Yoneda's lemma. Alternatively, just observe that polynomial functions are precisely ones commuting with every ring homomorphism.
In general, let P:C->B be a functor from an arbitrary C to a cartesian closed B. Select an object R in C and let R/P:R/C->B be the "obvious" forgetful functor from the co-slice category. For an object S of B define
R[S] = Nat(R/P * hom(S, -), R/P),
where hom is the internal hom functor of B and * is functor composition in the diagrammatical order. For C = Ring, B = Set this gives the usual ring of polynomials with coefficients in R and variables from S.
This construction extends to a functor C x B -> Set and has some nice properties of the usual polynomials: polynomials of "one variable" (i.e., when S = 1) can be composed, to each r:1->R and x:1->S corresponds a polynomial (provided P maps 1 to 1).
Has this or dual (where C/R is used instead of R/C) construction been studied? Maybe in enriched contexts?
Thank you, Nikita.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Nikita, The constructions that you describe can be understood as variations on Lawvere's concept of the _algebraic_structure_ of a (Set-valued) functor: F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, New York, 1963. Available in: Repr. Theory Appl. Categ. 5 (2004). Chapter III, Section 1. The first construction that you describe is an instance of Lawvere's notion, wherein the algebraic structure of the functor P is the algebraic theory T of commutative rings. The hom-set Nat(P,P) ~= Z[X] that you describe is one of the hom-sets of this algebraic theory T, namely the set of unary operations, which underlies the free T-algebra on one generator. In the case where we take B = Set, your second construction is an instance of Linton's formulation of the _equational_structure_ of a Set-valued functor as defined in F. E. J. Linton, Some aspects of equational categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, 1966, pp. 84-94. Section 2. This notion of equational structure is an adaptation of Lawvere's notion to the setting of Linton's _equational_theories_, which are algebraic theories in which the arities are allowed to be arbitrary (small) sets rather than just finite cardinals. Indeed, take U to be the relevant functor R/C --> B = Set and consider the equational structure T of U in Linton's sense. Given a set S, the set of natural transformations that you describe is one of the hom-sets of the resulting equational theory T, namely the (possibly large) set of all S-ary operations, which (if it is small) underlies the free T-algebra on the set S. In the general case, where B is an arbitrary cartesian closed category, your second construction is closely related to (but not an instance of) Dubuc's formulation of the algebraic structure of a (tractable) V-valued V-functor, as defined in E. J. Dubuc, Enriched semantics-structure (meta) adjointness, Rev. Un. Mat. Argentina 25 (1970), 5-26. Section 4. This notion of enriched algebraic structure is a generalization of Linton's above notion of equational structure (and so in turn is an adaptation of Lawvere's notion) to the context of the V-theories of Dubuc, which are V-enriched algebraic theories in which the arities are arbitrary objects of V (so that in the case V = Set, Linton's notion is recovered). As Dubuc demonstrates, this notion of algebraic structure is very closely related to the notion of codensity monad. The divergence between your second construction and Dubuc's formulation lies in the fact that you use an ordinary, non-enriched V=B-valued functor rather than a V-functor and, correspondingly, you form a set of natural transformations rather than an object of V-natural transformations. Note that above we have made no special use of the coslice category R/C, so this aspect can be analyzed separately. Best wishes, Rory -----Original Message----- From: Nikita Danilov Sent: Monday, April 04, 2016 10:32 AM To: categories@mta.ca Subject: categories: A construction for polynomials. Dear list, I am looking for a reference to the following construction. In the simplest case, consider the forgetful functor P:Ring->Set, from the category of commutative rings with a unit. Then the set of natural endomorphisms Nat(P, P) can be identified with the set of polynomials of one variable with integer coefficients Z[X]. This can be easily seen by the chain of isomorphisms Nat(P, P) = Nat(Ring(Z[X], -), P) = Z[X], where the first isomorphism is due to Z[X] being a free ring with one generator and the second is by Yoneda's lemma. Alternatively, just observe that polynomial functions are precisely ones commuting with every ring homomorphism. In general, let P:C->B be a functor from an arbitrary C to a cartesian closed B. Select an object R in C and let R/P:R/C->B be the "obvious" forgetful functor from the co-slice category. For an object S of B define R[S] = Nat(R/P * hom(S, -), R/P), where hom is the internal hom functor of B and * is functor composition in the diagrammatical order. For C = Ring, B = Set this gives the usual ring of polynomials with coefficients in R and variables from S. This construction extends to a functor C x B -> Set and has some nice properties of the usual polynomials: polynomials of "one variable" (i.e., when S = 1) can be composed, to each r:1->R and x:1->S corresponds a polynomial (provided P maps 1 to 1). Has this or dual (where C/R is used instead of R/C) construction been studied? Maybe in enriched contexts? Thank you, Nikita. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Dear Nikita, the special significant case with which you begin, deals with the forgetful functor (which you call P) from rings to sets. It lives in a category (topos) E of covariant functors from rings to sets, and this category you do not give a name. But P and E seem to me to be the main actors in your construction. What you call R[S] (for R in Rings, and S in Sets) is the value at R of the exponent object P^S -> P in E. In particular, R[1] is the value at R of the object P -> P in E. What you observe has as a special case the fact that P -> P, as an object in E, has the universal property of "P[X]", the free P-algebra in one generator; i.e. P[X] = (P -> P) in E. This coincidence, of a colimit type universal property (of P[X]), with a limit type universal property (of the exponential P -> P), is significant, and generalizes to further duality results in E. It also generalizes to any othe algebraic theory T, not just to the theory of rings. For instance, for T the initial algebraic theory (the algebraic theory of sets), it implies that P+1 = (P -> P). For an elaboration of these generalizations, see my "Duality for Generic Algebras", Cahiers 56 (2015), 2-14, or http://home.math.au.dk/kock/DGA03.pdf Anders [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Anders Kock -
Fred E.J. Linton -
Nikita Danilov -
Rory Lucyshyn-Wright